ATICAN CITY - For victims and their families, as well as Catholic priests and congregations outraged by what they saw as Cardinal Bernard F. Law's failure to protect children from sexually abusive priests, the decision was a long time coming.

But when Pope John Paul II accepted Law's resignation yesterday, the Catholic Church hierarchy showed unusual speed and decisiveness in addressing the sexual abuse crisis that erupted in January in Law's Boston archdiocese, Vatican observers say.

For an institution that measures time in centuries, not network news cycles, Vatican insiders said it was an extraordinarily swift and forceful response and a profound recognition that the Vatican had finally come to understand the depth of the crisis in the American church.

Yesterday's resignation was also a dramatic departure for the Vatican, an institution rarely swayed by public outcry or media scrutiny. It was unusual, too, for the pope to recognize the failings of a cardinal rather than encourage him to persevere in God's work and try to fix what was broken.

''Whatever desire there was to have Cardinal Law tough it out, to stay with the job, was washed away during a week in which it had become painfully clear that it was no longer possible for him to govern,'' said a senior Vatican official. None of us have really absorbed this yet. It will take a very long time, and there is much to be learned from it.''

Law arrived in Rome on Sunday, and throughout the week news kept breaking on the scandal and pressure kept mounting for him to resign. But the critical development that reverberated in the Vatican's corridors was the letter signed by 58 priests in the Boston archdiocese asking Law to resign, officials and observers said yesterday.

Law held a series of meetings with the Curia officials, including the prefect of the Congregation for the Clergy, Cardinal Dario Castrillon Hoyos, and with the prefect for the Congregation for Bishops, Cardinal Giovanni Battista Re.

Vatican sources said yesterday that Re and Castrillon were hesitant to accept Law's resignation, worried that it would be seen as the church bowing to public pressure. They feared a possible domino effect, since other cardinals in America, as well as Ireland and England, are also alleged to have failed to protect victims from abusive priests.

''It is very difficult for Rome to understand American Roman Catholics,'' said Andrew Maloney, a Vatican historian based in Rome. ''They can't see how this scandal revealed the spirit of public participation by American Catholics, and their sense of accountability.''

Some Vatican insiders said that Bishop James Harvey, an American cleric who is the head of the papal household, could have played a key role in helping the pope to understand the depth of the scandal engulfing Law and the Boston archdiocese. During his time in Rome this week, Law has stayed in Harvey's residence.

''For the pope to have finally understood what was going on in America would have required someone to translate, and I think it is quite possible that person was Harvey,'' said one Vatican source, who pointed out that Harvey is one of very few people who have daily access to the pope.

''There was strong resistance in the Vatican to take such a step,'' said Marco Politi, a veteran Vatican reporter for the Italian daily newspaper La Repubblica. ''But at this level, it was the choice of the pope, and it is clear he wanted to send strong signal that the Holy See understands the need to open a new page and to clean the table, as we say in Italy.''

John Allen Jr., the Vatican correspondent for the National Catholic Reporter, said yesterday's episode ''cut against the Vatican's institutional sensibility and the pope's personal psychology. So it underscores the uniqueness of what had happened in Boston and the depth of the scandal. It shows that it really was unlike any other scandal which the church has encountered.''